Guido,
I don't know if I can be much help. I do have a HP 334A although it is a different option that has all push buttons instead of the front panel dials yours has. But I think it operates the same otherwise. I don't use it very often since I also have a newer HP 339A.
On the voltmeter, if I select the 0.0003V range, with no input mine reads almost full scale (290 uV), but that is with the input open. With the input terminals shorted, it reads about 15 uV. With a 600 Ohm metal film resistor across the input terminals, it also reads about 15 uV.
The automatic nulling problem may require some troubleshooting. I have never had to work on mine, so I am not very familiar with it. But I understand it uses a pair of incandescent lamps to null the bridge that is part of the notch filter. So maybe a lamp has failed. The service manuals for most of the old HP equipment is available on the Agilent web site. The manual can be downloaded here:http://cp.literature.agilent.com/lit...0333-90008.pdf

The HP manuals are very good with block diagrams and troubleshooting guides.

The jitter on the voltmeter you see makes me think that maybe a filter capacitor has failed on one of the power supplies. That is pretty common in such old equipment. So a good place to start would be to measure each of the power supplies and also check them for excessive ripple.

Let me know if I can be of any further help. Good luck with fixing it.